This notation for the sum of vector spaces means that for any \(v_h \in V^k_{h,p}\) ,
there exist \(\lbrace v_i \in V_i\rbrace _i\)  such that \(v_h = \sum _i v_i\) . The decomposition
is not typically unique. Given an initial guess for the solution to a
variational problem posed over \(V^k_{h,p}\) , the Galerkin projection of the equation
for the error is solved over each \(V_i\)  (additively or multiplicatively). This
gives an approximation to the error in each subspace \(V_i\) , which are combined.
A cycle over each subspace constitutes one step of a subspace correction
method. For more details, see Xu [1]}.
